Hotspots of the stokes rotating circulation in a large marginal sea

Marginal seas, surrounded by continents with dense populations, are vulnerable and have a quick response to climate change effects. The seas typically have alternatively rotating layered circulations to regulate regional heat and biogeochemical transports. The circulations are composed of dynamically active hotspots and governed by the couplings between unique extrinsic inflow and intrinsic dynamic response. Ambiguities about the circulations’ structure, composition, and physics still exist, and these ambiguities have led to poor numerical simulation of the marginal sea in global models. The South China Sea is an outstanding example of a marginal sea that has this typical rotating circulation. Our study demonstrates that the rotating circulation is structured by energetic hotspots with large vorticity arising from unique dynamics in the marginal sea and is identifiable by the constraints of Stokes Theorem. These hotspots contribute most of the vorticity and most of energy needed to form and maintain the rotating circulation pattern. Our findings provide new insights on the distinguishing features of the rotating circulation and the dominant physics with the objectives of advancing our knowledge and improving modeling of marginal seas.

G lobal marginal seas are separated from the major ocean basins by topographic features and links to the adjacent oceans such as the Atlantic Ocean (e.g., Baltic Sea 1 , Mediterranean Sea 2 , Gulf of Mexico 3 ), Indian Ocean (e.g., Red Sea 4 ), or Pacific Ocean (South China Sea, SCS 5 ). These marginal seas exhibit rotating circulation patterns that are forced by an inflow-outflow water exchange with the adjacent oceans through straits such as Skagerrak Strait (connecting to Baltic Sea), Gibraltar Strait (Mediterranean Sea), Florida Strait (Gulf of Mexico), Bab-al-Mandeb Strait (Red Sea), and Luzon Strait (SCS). Other forcings include the local wind forcing and the coupled internal dynamics processes mainly governed by local topography and thermohaline structures [6][7][8][9][10] . These marginal seas generally have a deep basin surrounded by a steep slope, and their circulation are regulated by coupled external-internal dynamics. To demonstrate and explain the dynamics, we use as our example of a typical marginal sea, the SCS, which is the largest marginal sea in the Southeast Asia (Fig. 1a) and is surrounded by countries with~22% of the world's total population based on the data of World Population Prospects 2019 from United Nations (https:// population.un.org/wpp/).
Identifying the rotating layered circulation was initially based on estimates and speculation for only parts of the basin, at certain depths, or across specific transects. The circulation was represented with respect to geopotential [12][13][14] or isopycnal 15,16 levels. Under the wind stress curl forcing and due to the path of the SCS throughflow 20,21 , the CAC circulation in the SCS consists of complex small-scale circulations that are regulated by local dynamics. Furthermore, the CAC circulation's existence is based only on domain integration of vorticity, which is formulated as the Stokes' Circulation Theorem (see Eq. (1) in Methods section), over the entire basin and relative to geopotential levels ( Fig. 1b-d). We found that the layered CAC circulation does not exist in an isopycnal coordinate system (Fig. 1d) where the Stokes' circulation has a different physical meaning with when the circulation is viewed at the geopotential levels 22,23 . Our results show that the difference of circulations defined in geopotential and isopycnal coordinates is particularly significant over the continental slope where change of isopycnal depth is dramatic. Similar problems in depicting the circulation also exist in other marginal seas [1][2][3][4][5] .
It is the layered influx-outflux-influx of planetary vorticity through the Luzon Strait (LS, Fig. 1a) and the corresponding vortex squeezing or stretching in the SCS basin that cause the CAC circulation to form in the basin 10 . Although the coupled internal-external dynamics have recently been found to link to the vertical transport associated with the cross-slope motion and three-dimensional energy transport and transfer 23,24 , the detailed dynamics that frame the CAC circulation in the SCS remains unknown.
We validated CAC circulation in the SCS with geostrophic currents derived from GDEM (Fig. 1c) and with both observations and geophysical fluid dynamics reasoning 10,25 . We also found that most of the global circulation models with the same or higher resolutions, such as CMEMS 26 , OFES 27 , HYCOM 28 , and others (not shown) have not well captured the layered CAC circulation in the SCS (Fig. 1c). These global models failed to simulate the exchanges between the SCS and the adjacent oceans. Yet these exchanges provided crucial external fluxes to couple with internal dynamics for the formation of the CAC circulation. A similar modeling problem has also been reported for other marginal seas [29][30][31] . There are many reasons for this deficiency in the global model. A better understanding and thus better representation of circulation physics in the complex marginal sea such as flow-topography interaction internally and strait dynamics externally are certainly critical to improving of regional simulations in the global models, and it is the aim of this study to provide both.

Results
In the SCS, the CAC circulation is strongly spatially variable and contributed mainly by vorticities (ζ) in several key regions of the basin according to Stokes' Circulation Theorem (Fig. 2a). To distinguish these key regions of the CAC circulation from the other circulation, we defined a favorable vorticity (ζ f ) that creates the CAC circulation for which ζ f > 0 in the upper and deep layers, and ζ f <0 in the middle layer. Figure 2b shows regions with relatively larger ζ f that we recognized as hotspots for the CAC circulation. These hotspots are distinguished by ζ f that is greater than ζ f averaged over the entire domain for each layer. In this study, the hotspots are bounded by ζ f >2:2 10 À6 m s −1 , ζ f <−1.1 10 À6 m s −1 , and ζ f >0:91 10 À6 m s −1 in the upper, middle, and deep layers, respectively. In the end, we identified four distinct hotspots with strong ζ f . These hotspots are along the continental slope (SLO), off Luzon Strait (LS), off Mindoro Strait (MS), and in the southwestern basin (SWB) (Fig. 2b). Vorticity in these hotspots spatially matches the net vorticity averaged over each layer in Fig. 2a, which indicates that the hotspots dominantly contribute most of the vorticity that forms the CAC circulation. Figure 3a shows that the total ζ f contributions of IR vort (see Eq. (2) in Methods section), from the hotspots to the basin CAC circulation are 58%, 55%, and 49% with the corresponding areas of 35.2%, 31.0%, and 33.3% in the upper, middle, and deep layers, respectively. Most of the contributions are from SLO ( Fig. 2b) and the contribution along the western SLO to the CAC circulation is prominent (Fig. 2a). Thus, although the SLO occupies~28.5%, 23.2%, and 19.1% of the basin area, the slope current flowing along the curved slope region has IR vort of 47%, 39%, and 23% in the upper, middle, and deep layers, respectively. Nevertheless, even though IR vort is relatively small for LS, MS, and SWB, because of the areas they occupy arẽ 4.6%, 2.6%, and 7.1% of the basin area, respectively, these hotspots are still dynamically active and important to the CAC circulation, as shown by their contributions per area or AR VORT values.
In addition to have a relatively large vorticity contribution to CAC as shown by IR VORT values, the hotspots are also characterized with active dynamics measured by AR VORT (see Eq. (3) in Methods section). AR vort reflects the intensity of the active dynamics or the contributions of ζ f per unit area to the CAC circulation in each hotspot. In Figs. 2 and 3, all AR vort > 1, which means the intensity of ζ f for each hotspot is much stronger than for the rest of the basin. Unlike IR vort , AR vort values in LS, MS, and SWB ( Fig. 2b) are much larger than that in SLO, suggesting that the active dynamics in the CAC circulation are induced either by the exchanging flows between the SCS and the adjacent oceans through LS and MS or by flow over a topographic trough in SWB, respectively. ζ f in the hotspots is induced either by horizontal velocity shear in the jet-like slope current or by the curvature vorticity regulated   by the curving isobaths along SLO. Figure 3a indicates that it is the shear vorticity that dominates the total vorticity in the hotspots. Our results ( Supplementary Fig. 1) show that the relatively large contributions of shear vorticity are mainly found on the seaward side of the slope current along SLO, and the curving slope topography's contribution is less important contribution to ζ f . In addition, we found that the vorticity and dynamics that the hotspots contribute persist throughout the year even though the intensity of the CAC circulation is highly variable in time (Fig. 1b).

Discussion
We investigated the vorticity balance in the hotspots to gain insight into the physics that determine how the CAC circulation forms in the SCS. Figure 4a shows the terms of the vorticity equations (see Eqs. (4) and (5) in Methods section) normalized within the hotspots. Ω cor in all three layers predominantly contributes the vorticity to all the hotspots, which is indicated by Ω cor >0 in the upper and deep layers and Ω cor <0 in the middle layer. Ω cor is related intrinsically to the stretching or squeezing of the water column along SL. In addition, Ω cor extrinsically linked the influx of planetary vorticity (see Eq. (6) in Methods) into SLO in the upper and deep layers and the outflux in the middle layer 10 . Our analyses (Supplementary Fig. 2) showed that an additional contribution of positive Ω adv in the upper layer forms in LS because of the westward influx of positive relative vorticity on the western side of the Kuroshio jet and because there is a larger kinetic energy (KE) gradient 22 (see Eq. (7) in Methods section). We see that Ω adv is small in the other layers.
The balance between the bottom pressure torque Ω pgf and Ω cor in the hotspots reflects the intrinsic dynamics of the flow interacting with topography in response to the external influx or outflux across the boundaries of the hotspots. In the deep layer, the major sinks for the positive Ω cor are nonlinear advection ðΩ adv Þ and bottom stress curl ðΩ b Þ. A larger negative Ω adv occurs in the deep layers of SWB, MS, and LS where an abruptly changing bottom topography forms stronger nonlinearities in the flow. Ω b is negative in all layers and is relatively large in the upper and deep layers because of a stronger current and topographic influences, respectively.
The vorticity balance and underlying physics in the hotspots generally agree with the vorticity balance and physics for the entire basin 10 , as shown by the dashed lines in Fig. 4a, which confirms the dominant role of the hotspots in forming the CAC circulation. However, some unique dynamics occurring within the hotspots deviate from the conditions occurring in the whole basin. For example, over an annual timescale, the wind stress curl imparts a net positive ζ f for the cyclonic circulation in the upper layer over the entire basin 10 . However, in SLO where the IR vort is the largest, the negative wind stress curl in the northern basin during the winter balances the positive wind stress curl in the southern basin in the summer, and the annual net effect of wind stress curl on ζ f in the hotspots is negligible. We identified that the net wind stress curl contributing to the annual mean CAC circulation comes mainly from the west of the Luzon islands  during winter. Another unique hotspot dynamics phenomenon is the presence of positive Ω pgf in the deep layers of the hotspots. Our analyses (Supplementary Fig. 3) showed that the Ω pgf >0 is induced by topographic troughs in SWB and LS where a relatively strong positive Ω pgf occurs as flows climb up the troughs and squeeze the water column to balance the negative Ω cor . This finding suggests that through the follow-topography interaction, topography in a marginal sea can play an active, rather than just passive only, role in forming the CAC circulation.
The energy balance provides insight into how the hotspots contribute energy to the CAC circulation. The mean kinetic energy (MKE) from the hotspots reflects the strength of the mean circulation that maintains the CAC circulation while the terms of the MKE equation (see Eqs. (9) and (10) in Methods section) provides the associated energy source and sink. Like the vorticity contribution from the hotspots, there is a large amount of MKE in these hotspots in all layers 20 , which further points to the active dynamic role of the hotspots have in the CAC circulation. Figure 3b shows that in the upper layer, the hotspots substantially contribute MKE (IR M = 81%) to the CAC circulation (Fig. 3b). Most of the MKE (76%) occurs along SLO (Fig. 2a), indicating that the slope current primarily sustains the CAC circulation. The averaged AR M for the hotspots is 2.21 times larger than the average for the entire basin (Fig. 3b), ranging from 0.7 in MS to 2.5 in SLO (Fig. 2a).
In the upper layer, the major contributors to the MKE are WP M and DIV M (Fig. 4b). DIV M primarily occurs in LS in the upper layer, because of the kinetic energy influx from the Kuroshio intrusion. Major WP M exists in SLO in the upper layer and indicates the dominance of wind power input in driving the slope current. Unlike the small net wind stress curl contribution to the vorticity balance along SLO (Fig. 4a), the wind power input is the major kinetic energy source that drives and amplifies the slope current component of the CAC circulation. Wind power explains why the slope current in the SCS is 3 to 4 times greater than the volume of the Kuroshio intrusion through LS (~5.9 Sv, Fig. 1a) 25 . The strengthened slope current also enhances the basin-wide recirculation for maintaining the CAC circulation. MKE in the upper layer is dissipated by VIS M and converted to eddy kinetic energy by BTI M .
In the middle layer, there is a smaller, but still significant IR M (~68%) (Fig. 3b) in the hotsots, where SLO accounts for~43% of the total MKE, and LS and MS, combined, have~25% of the MKE (Fig. 2a). Contributions of AR M are up to 2.32 times larger than the basin average in the middle layer, with very large contributions from MS and SLO: 5.73 times larger than the basin average in MS and 1.90 times larger than the basin average in SLO (Fig. 2a). Furthermore, along SLO and near MS and LS in the middle layer, mean pressure work PW M mainly induces MKE and is balanced by barotropic BTI M , baroclinic instability BCI M, and VIS M (Fig. 4b). These results indicated that the extrinsic pressure flux and the intrinsic dynamic response are the major source and sink for the hotspot energy in the middle layer, respectively.
In the deep layer, although IR M is the smallest due to the relatively small geographic sizes of the hotspots, the hotspots are dynamically active enough to contribute 50% (IR M = 0.5) to the CAC circulation in the basin. Interestingly, there is a larger contribution of energy in SWB where there is topographic trough (Fig. 3b). The mean AR M of the hotspots is 1.49, ranging from a very strong AR M of~4.72 in SWB to a relatively small value of 0.69 in SLO (Fig. 2a).
The origin of MKE in the hotspots of the deep layer is primarily from the strong circular current and a meandering current in SWB and SLO, respectively. The major source of the MKE in this layer is generated by BCI M (Fig. 4b) or by potential energy released due to the vertical motion over the trough. The important role of topography is obvious in the deep layer, as is also shown by the larger Ω pgf in SWB (Fig. 4a).
EKE is also distinguished in hotspots, with significant contributions of IR E (0.53, 0.47, 0.43) and AR E (1.46, 1.60, 1.28) in different (upper, middle, deep) layers (Fig. 3b). The large values of IR E in all layers occur mainly along SLO, while the active local dynamics in other hotspots also generate significant AR E (Fig. 2a). Besides input by wind power WP E in the upper layer, EKE is mainly contributed by internal pressure work and conversion from MKE through internal BCI E due to active dynamics of the hotspots in both middle and deep layers (Fig. 4c). The distinguished EKE in the hotspots shows their important contribution to the variability of the basin circulation.
This study shows the active dynamics of the hotspots that govern the characteristics, composition, and formation of the CAC circulation, and clarifies ambiguities regarding the physics of the CAC circulation in the SCS. The findings demonstrate an innovative concept for diagnosing the dominant physics of layered rotating circulation in marginal seas. By illustrating the importance of internal hotspots dynamics, and thus the coupled internal-external mechanism, we present the challenge in improving our physics comprehension and the associated modeling in the marginal seas.

Methods
In our study, we used the China Sea Multi-scale Ocean Modeling System (CMOMS, https://odmp.ust.hk/cmoms/) 10 CMOMS is dynamically configurated and numerically implemented in the SCS, and the model results are observationally and physically well-validated 10,25 . For our analyses, we used 20-year model variables from 1992 to 2011 that were averaged over 7 days at synoptic-scales.
Stokes' Circulation Theorem for the entire basin relative to geopotential levels is mathematically represented as where Γ is the basin circulation integrated along the boundary of the SCS basin with area A SCS ;Ṽ H is horizontal velocity vector; and ζ=∇ H ×Ṽ H is the vertical relative vorticity normal to A SCS . ζ f and the mean kinetic energy (MKE) contributions from our identified hotspots to the CAC circulation are measured by the relative contribution IR ðvort;MÞ , and by the intensity (contribution per unit area) of the active dynamics within the hotspots which are represented AR (vort,M) , These parameters determine the dynamic characteristics of the hotspots in the CAC circulation.
We expressed the vorticity equation, integrated by volume (V) over a specific layer in the hotspots, as 8 , where and Ω adv ¼ À where ζ t is volume-integrated changing rate of vorticity. The terms Lu and Lb are the depths, and τ u and τ b are the stress at the top and bottom for each specific layer, respectively. J is a Jacobean operator and NL is the nonlinear advection. P Lu and P Lb are pressure at the top and bottom of each layer, respectively, and D is the total water depth. S i and S j are are the meridionally and zonally oriented sections., which are positive outward of the enclosed SCS area. The first term on the right-hand side of Eq. (4) is the layer-integrated divergence Ω cor or (vortex stretching/squeezing), which is the equivalent to planetary vorticity flux across the open boundary of the hotspot region 10 . Ω adv is the sum of horizontal relative vorticity advection and tilting, where horizontal advection represents the combined effects of relative vorticity flux and the kinetic energy gradient across the boundary. Ω pgf is the bottom pressure torque reflecting the interaction between the baroclinically induced pressure and the variable slope topography, and Ω u and Ω b represent upper and bottom stress curl in each specific layer, respectively. These stress curls specifically refer to wind stress and bottom frictional stress curls in the upper and deep layers, respectively. Ω h represents horizontal surface (v u t ) and bottom (v b t ) velocity accelerations interacting with the respective surface elevation ðη u Þ and bottom topographyðη b Þ. We found that Ω h is generally small and can be neglected. Equation (4) explicitly illustrates circulation physics for a semi-closed marginal sea such as the SCS by distinguishing the coupled intrinsic and extrinsic dynamics for the CAC circulation.
We define MKE and EKE as 24 where ( u, v) are the time mean of the horizontal velocity and (u 0 , v 0 ) indicate their eddy states.
The subscript, M and E refers to MKE and EKE, respectively, and Eq. (9) is fully expressed as 20 Àgρ 0 a w 0 |fflfflffl ffl{zfflfflffl ffl}